Random search in a multi-target environment

Foraging, rescue operations, environmental mapping and disease spread are examples where animals, humans, robots or pathogens aim to find either of a set of potential targets in space, being their static or mobile. In the absence of cues from the environment a random search may represent an effective way to find such targets. When the environment is discrete and confined, past studies has focused mainly on estimating the mean first-passage time of a random walker to one or a small set of targets. The lack of an exact analytic expression for the spatial occupation probability, or propagator, of a discrete random walker has represented one of the limitations to develop a general theory of multi-target search in confined space. Here I present such a theory by deriving the exact propagator, obtaining expressions for the first-passage probability and the mean first-passage time to any number of targets, as well as the encounter and transmission probability of pairs of walkers in any dimensions.